\section{Monte Carlo}
To approximate the value the stock price numerically several approaches can be
used. In this section two methods will be considered. First the approximations
will be derived. This is followed by a discussion of the random paths
generated using these approximations. It will be showed that by increasing the
number of approximations the error can be reduced.

\subsection{Numerical Approximations of a Stock Price}
In the previous section we have argued that a stock price follows a log normal
distribution. Therefore, the stock price can be considered as a random walk.
To deal with this probabilistic behaviour of the price specific numerical
methods have to be used. The prevailing approach for this is the use of Monte
Carlo techniques. Next two Monte Carlo algorithms will be discussed.

The first algorithm is based on the deterministic forward Euler method. The
basic Euler approximation is defined as:
\[
	S_n' = \frac{S_{n + 1} - S_n}{h}
\]
The terms can be rearranged to $S_{n + 1} = S_n + hS_n'$. This deterministic
equation can be extended to a probabilistic by substituting $S_n'$ with its
numerical approximation: $S_n' = S_n r \Delta t + S_n \sigma \Delta W$. Then
by setting $h = 1$ the Euler approximation for a stock price is obtained:
\begin{equation}\label{eq:euler}
S_{n + 1} = S_n + S_n (r \Delta t + \sigma\Delta W)
\end{equation}
Since the variance of $\Delta W$ is $\Delta t$, $\Delta W$ can be generated
using $Z\sqrt{\Delta t}$ with $Z \sim N(0, 1)$.\\

A second approximation uses the assumption that a stock price is log normally
distributed with a drift term $\mu = (r + \sigma^2 / 2)$ and a diffusion term
$\sigma$. The SDE for this process is given in equation \eqref{eq:sde}.
Recall that the integrated form is of the form:
\begin{equation}\label{eq:int}
S(t)=S(0)e^{(r-\frac{\sigma^2}{2})t+\sigma Z\sqrt t}.
\end{equation}

\subsection{Monte Carlo method}
A Monte Carlo simulation is performed in the following way:
\begin{enumerate}
	\item simulate 1 path for the stock price in a risk neutral world
	\item Calculate the payoff from the stock option
	\item Repeat steps 1 and 2 many times to get a sample payoff
	\item Calculate mean payoff
	\item Discount mean payoff at risk free rate to get an estimate of the value of an option.
\end{enumerate}
Due to the `Law of large numbers' the Monte Carlo estimate converges to the
true value, and due to the `Central limit theorem' the Monte Carlo estimate is
asymptotically normally distributed. If we call $M$ the number of times steps
1 and 2 are repeated then the standard deviation of the Monte Carlo estimate
is given by $\frac{\sigma(payoff)}{\sqrt N}$. With `payoff' the premium of an
option is meant, e.g the payoff of an European call option is $(S_T-K)^+$.\\

We have priced European call and put options with ($T=1$ year, $K=99$, $r=
6\%$, $S=100$ and $\sigma=20\%$) for both approximations with the Monte Carlo
method. For the first step the formulas $S_{n + 1} = S_n + S_n (r \Delta t +
\sigma Z\sqrt{\Delta t}$ and $S(t)=S(0)e^{(r-\frac{\sigma^2}{2})t+\sigma Z
\sqrt t}$ were used. Some realizations are showed in \ref{fig:randomwalks}.

\begin{figure*}[h]
\caption{Realizations of the stock price under both Monte Carlo
approximations.}\label{fig:randomwalks}
\begin{center}
	\includegraphics[width=\textwidth]{random_walks}
\begin{minipage}{\textwidth}
In the figure 20 realizations of a random walk using equation \eqref{eq:euler}
and \eqref{eq:int} are plotted. The red and blue lines are the mean of the
random walks. The parameters of the equations are equal the same as throughout
this paper..
\end{minipage}
\end{center}
\end{figure*}

For the second step the payoffs were calculated: $(S_T-K)^+$ and $(K-S_T)^+$
for the call and put option respectively. This was repeated $M$ times and
step 4 and 5 were merged to give the following value for the option price.
\[
V(S(0))=e^{-rt}\frac{\sum_{m=1}^M payoff^m(S^N)}{M}
\]
Here $N=\frac{T}{\delta t}$ and $S^n=S(n\delta t)$, the discretization of the
stock price.



\subsection{Convergence studies}\label{sec:conver}
The values that were obtained with the Monte Carlo method were compared with
the results from the analytical value.

\begin{figure*}[h]
\caption{Convergence Monte Carlo to analytical value.}\label{fig:approximationdiff}
\begin{center}
	\includegraphics[width=14cm]{approximation_diff}
\begin{minipage}{\textwidth}
{\footnotesize When more trials are used the Monte Carlo approximations
converge better to the analytical value, many trials are needed to get a rough
approximation of the analytical value.  }
\end{minipage}
\end{center}
\end{figure*}

In figure \ref{fig:approximationdiff} it can be seen that the variance stays
rather big although many realisations were performed. In order to get a
reduced variance we need to use a sophisticated method.

One method to do this is the `Antithetic variable technique'. Both the
approximations we used were highly dependent of the random numbers that were
drawn in that trial so if we want to reduce the variance we need to reduce the
variance of those random numbers. 
Recall $S_{n + 1} = S_n + S_n (r \Delta t + \sigma Z\sqrt{\Delta t})$ and
$S(t)=S(0)e^{(r-\frac{\sigma^2}{2})t+\sigma Z \sqrt t}$ with $Z \sim N(0,1)$.
Suppose we calculate the value of an option in the following way $\tilde
V=\frac{V^+ + V^-}{2}$ with $V^+$ the Monte Carlo estimated based on $Z^n$ and
$V^-$ the Monte Carlo estimate based on $-Z^n$. Then the variance calculated
in this way, $\tilde V$, is less then the original value, $V$, this is shown
in the following equation.
\[
Var(\tilde V)=\frac{1}{4} Var(V^+)+\frac{1}{4}Var(V^-)+\frac{1}{2}Cov(V^+,V^-)
\]
Due to the correlation between $Z^n$ and $-Z^n$ the covariance becomes
negative resulting in a smaller variance.

This `Antithetic variable technique' was implemented in the Monte Carlo method
and again convergence studies were done, shown in figure
\ref{fig:approx}

\begin{figure*}[h]
\caption{Call option price estimation}\label{fig:approx}
\begin{center}
\subfloat[Antithetic Variance technique in call option]{%
	\label{fig:calloptionmc}\includegraphics[width=7cm]{call_option_mc}}\qquad
\subfloat[Absolute approximation error]{\label{fig:mc_approx_error}%
	\includegraphics[width=7cm]{mc_approx_error}}\\
\medskip
\begin{minipage}{14cm}
{\footnotesize The figures show that the Antithetic Variance reduction
technique results in a better approximation of the analytical value.
}
\end{minipage}
\end{center}
\end{figure*}

\subsection{Delta Hedge}\label{sec:delta}
As stated earlier the delta hedge, $\delta$, is the number of units of stock
one should hold for each option sorted in order to create a risk-less hedge.
This is equal to the change in option price divided by the change is stock for
each time period. For a short period of time this results in
\[
\delta = \frac{\delta V}{\delta S} =
\lim_{\varepsilon \rightarrow 0} \frac{V(S+\varepsilon)-V(S)}{\varepsilon}.
\]
The delta hedge was implemented for both approximations that were used in the
Monte Carlo method. The results of the approximation is showed in figure
\ref{fig:deltamc}.

\begin{figure*}[h]
\caption{Delta Hedge Approximation}\label{fig:deltamc}
\begin{center}
\subfloat[The delta hedge approximation]{%
	\includegraphics[width=7cm]{delta_mc}}\qquad
\subfloat[The absolute error in the approximation]{%
	\includegraphics[width=7cm]{delta_mc_error}}\\
\medskip
\begin{minipage}{14cm}
{\footnotesize The figures show that the variable reduction technique results
in a better approximation of the analytical value. We use $\varepsilon = 0.1$
}
\end{minipage}
\end{center}
\end{figure*}

\subsection{Digital option}
A digital option is an option which pays one euro if the stock price at expiry
is higher than the strike and otherwise nothing. From such an option we wanted
to calculate the hedge parameter, $\delta$. The price of a digital option can
be approximated with a short and a long call option. If payoff from the long
option is $(S_T-(K-\frac{1}{2}))^+$ and the payoff from the short call option
is $-(S_T-(K+\frac{1}{2})^+$, this results in the following payoff
\[
payoff=\left\{
\begin{array}{ll}
(S_T-(K-\frac{1}{2}))-(S_T-(k+\frac{1}{2}))=1&\mbox{if $S_T>K+\frac{1}{2}$};\\
S_T-(K-\frac{1}{2})&\mbox{if $K-\frac{1}{2}\leq S_T\leq K+\frac{1}{2}$};\\
0&\mbox{if $S_T<K-\frac{1}{2}$}.
\end{array}
\right.
\]

A figure of this approximation is \ref{fig:digital}
\begin{figure*}[h]
\caption{Digital option}\label{fig:digital}
\begin{center}
	\includegraphics[width=10cm]{digital_port}
\begin{minipage}{10cm}
{\footnotesize The digital option approximated with a short and a long call option.
}
\end{minipage}
\end{center}
\end{figure*}

The value of the option is calculated with Monte Carlo, in the same way as
described for the European call and put options. With this obtained value the
delta hedge is calculated in the same way as described in section
\ref{sec:delta}. These calculations are also done with the help of variance
reduction, as described in section \ref{sec:conver}. The results of the delta
hedge approximation is showed in figure \ref{fig:digdhmc}. For an $\varepsilon  <
0.1$ we find that the Antithetic Variance methods converge. However, when
$\varepsilon > 0.5$ the Antithetic Variance methods converge very slowly or even
not (with 15000 random walks).

\begin{figure*}[h]
\caption{Digital option Delta Hedge approximation}\label{fig:digdhmc}
\begin{center}
	\includegraphics[width=8cm]{dh_approx_digital}

\begin{minipage}{8cm}
{\footnotesize The delta hedge of a digital option is approximated using the
techniques discussed in the text. As with the other approximations we see
that the Antithetic Variance reduction leads to convergence.
}
\end{minipage}
\end{center}
\end{figure*}


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